Download A Complex Representation of Sinusoidal Functions

1/25/2011
A Complex Representation of Sinusoidal Functions.doc
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A Complex Representation
of Sinusoidal Functions
Q: So, you say (for example) if a linear two-port circuit is
driven by a sinusoidal source with arbitrary frequency ωo ,
then the output will be identically sinusoidal, only with a
different magnitude and relative phase.
C
+
+
v1(t ) =Vm 1 cos (ωot + ϕ1 )
L
v2(t ) =Vm 2 cos (ωot + ϕ2 )
R
−
−
How do we determine the unknown magnitude Vm 2 and phase ϕ2
of this output?
A: Say the input and output are related by the impulse
response g (t ) :
v2(t ) = L ⎡⎣v1(t )⎤⎦ =
t
∫ g (t − t ′) v (t ′) dt ′
1
−∞
We now know that if the input were instead:
v1 (t ) = e j ω t
0
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/25/2011
then:
A Complex Representation of Sinusoidal Functions.doc
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v2(t ) = L ⎡⎣e j ω t ⎤⎦ = G (ω 0 ) e j ω t
0
where:
0
∞
G (ω 0 ) ∫ g (t ) e − j ω t dt
0
0
Thus, we simply multiply the input v1 (t ) = e j ω t by the complex
eigen value G (ω 0 ) to determine the complex output v 2 (t ) :
0
v2(t ) = G (ω 0 ) e j ω t
0
Q: You professors drive me crazy with all this
math involving complex (i.e., real and imaginary)
voltage functions. In the lab I can only generate
and measure real-valued voltages and real-valued
voltage functions. Voltage is a real-valued,
physical parameter!
A: You are quite correct.
Voltage is a real-valued parameter, expressing electric
potential (in Joules) per unit charge (in Coulombs).
Q: So, all your complex formulations and complex eigen
values and complex eigen functions may all be sound
mathematical abstractions, but aren’t they worthless to us
electrical engineers who work in the “real” world (pun
intended)?
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/25/2011
A Complex Representation of Sinusoidal Functions.doc
3/8
A: Absolutely not! Complex analysis actually simplifies our
analysis of real-valued voltages and currents in linear circuits
(but only for linear circuits!).
The key relationship comes from Euler’s Identity:
e j ωt = cos ωt + j sin ωt
Meaning:
Re {e j ωt } = cos ωt
Now, consider a complex value C. We of course can write this
complex number in terms of it real and imaginary parts:
C =a + j b
∴ a = Re {C }
and
b = Im {C }
But, we can also write it in terms of its magnitude C and
phase ϕ !
C = C e jϕ
where:
C = C C ∗ = a 2 + b2
ϕ = tan −1 ⎡⎣b a ⎤⎦
Thus, the complex function C e j ω t is:
0
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/25/2011
A Complex Representation of Sinusoidal Functions.doc
4/8
C e jω t = C e jϕ e jω t
0
0
= C e j ω t +ϕ
0
= C cos (ω 0t + ϕ ) + j C sin (ω 0t + ϕ )
Therefore we find:
C cos (ω 0t + ϕ ) = Re {C e j ω t }
0
Now, consider again the real-valued voltage function:
v1(t ) =Vm 1 cos (ωt + ϕ1 )
This function is of course sinusoidal with a magnitude Vm 1 and
phase ϕ1 . Using what we have learned above, we can likewise
express this real function as:
v1(t ) =Vm 1 cos (ωt + ϕ1 )
= Re {V1 e j ωt }
where V1 is the complex number:
V1 = Vm 1 e j ϕ
1
Q: I see! A real-valued sinusoid has a magnitude and phase,
just like complex number. A single complex number (V ) can
be used to specify both of the fundamental (real-valued)
parameters of our sinusoid (Vm , ϕ ).
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/25/2011
A Complex Representation of Sinusoidal Functions.doc
5/8
What I don’t see is how this helps us in our circuit analysis.
After all:
v2(t ) ≠ G (ωo ) Re {V1 e j ωot }
What then is the real-valued output v2(t ) of our two-port
network when the input v1(t ) is the real-valued sinusoid:
v1(t ) =Vm 1 cos (ωot + ϕ1 )
= Re {V1 e j ωot }
???
A: Let’s go back to our original convolution integral:
v2(t ) =
t
∫ g (t − t ′) v (t ′) dt ′
1
−∞
If:
v1(t ) =Vm 1 cos (ωot + ϕ1 )
= Re {V1 e j ωot }
then:
v2(t ) =
t
∫
g (t − t ′ ) Re {V1 e j ωot ′ } dt ′
−∞
Now, since the impulse function g (t ) is real-valued (this is
really important!) it can be shown that:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/25/2011
A Complex Representation of Sinusoidal Functions.doc
v2(t ) =
t
∫ g (t − t ′) Re {V e
1
j ωot ′
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}dt ′
−∞
⎧t
⎫
′
= Re ⎨ ∫ g (t − t ′ )V1 e j ωot dt ′⎬
⎩ −∞
⎭
Now, applying what we have previously learned;
⎧t
⎫
v2(t ) = Re ⎨ ∫ g (t − t ′ )V1 e j ωot ′dt ′⎬
⎩ −∞
⎭
⎧ t
⎫
′
= Re ⎨V1 ∫ g (t − t ′ ) e j ωot dt ′⎬
⎩ −∞
⎭
= Re {V1 G (ω0 ) e j ωot }
Thus, we finally can conclude the real-valued output v2(t ) due
to the real-valued input:
v1(t ) =Vm 1 cos (ωot + ϕ1 )
= Re {V1 e j ωot }
is:
v2(t ) = Re {V2 e j ωot }
=Vm 2 cos (ωot + ϕ2 )
where:
V2 = G (ωo )V1
The really important result here is the last one!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/25/2011
A Complex Representation of Sinusoidal Functions.doc
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C
+
+
v1(t ) =Vm 1 cos (ωot + ϕ1 )
L
v2(t ) = Re {G (ωo )V1 e j ωot }
R
−
−
The magnitude and phase of the output sinusoid (expressed as
complex value V2 ) is related to the magnitude and phase of the
input sinusoid (expressed as complex value V1 ) by the system
eigen value G (ωo ) :
V2
= G (ωo )
V1
Therefore we find that really often in electrical engineering,
we:
1. Use sinusoidal (i.e., eigen function) sources.
2. Express the voltages and currents created by these
sources as complex values (i.e., not as real functions of
time)!
For example, we might say “ V3 = 2.0 ”, meaning:
V3 = 2.0 = 2.0 e j 0
Jim Stiles
{
}
⇒ v3 (t ) = Re 2.0 e j 0e j ωot = 2.0 cos ωot
The Univ. of Kansas
Dept. of EECS
1/25/2011
A Complex Representation of Sinusoidal Functions.doc
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Or “ I L = −3.0 ”, meaning:
I L = −2.0 = 3.0 e j π
⇒
iL (t ) = Re {3.0 e j π e j ωot } = 3.0 cos (ωot + π )
Or “Vs = j ”, meaning:
Vs = j = 1.0 e
( )
j π2
{
⇒ v s (t ) = Re 1.0 e
}
(
( ) j ωot
e
= 1.0 cos ωot + π 2
j π2
)
* Remember, if a linear circuit is excited by a sinusoid (e.g.,
eigen function exp ⎡⎣ j ω 0t ⎤⎦), then the only unknowns are
the magnitude and phase of the sinusoidal currents and
voltages associated with each element of the circuit.
* These unknowns are completely described by complex
values, as complex values likewise have a magnitude and
phase.
* We can always “recover” the real-valued voltage or
current function by multiplying the complex value by
exp ⎡⎣ j ω 0t ⎤⎦ and then taking the real part, but typically we
don’t—after all, no new or unknown information is
revealed by this operation!
+
V1
C
+
L
R
−
Jim Stiles
V2 = G (ωo )V1
−
The Univ. of Kansas
Dept. of EECS